Dr. Marcus R Garvie

Published papers:


  1. Blowey J.F. and Garvie M.R. (2005), "A reaction-diffusion system of lambda-omega type. Part I: mathematical analysis," published in European Journal of Applied Mathematics, Vol. 16 No. 01 , pp. 1-19

    We rigorously establish the well-posedness of the strong solutions for an important class of oscillatory reaction-diffusion systems containing a supercritical Hopf bifurcation in the reaction kinetics. This is achieved using the classical Faedo-Galerkin method of Lions and compactness arguments. Furthermore, we present a complete case study for the application of this method to systems of reaction-diffusion equations.



  2. Garvie M.R. and Blowey J.F. (2005), "A reaction-diffusion system of lambda-omega type. Part II: numerical analysis," published in European Journal of Applied Mathematics, Vol. 16 No. 05 , pp. 621-646

    In this paper the results from the Part I paper are mimicked in the discrete case to present results for a fully-practical piecewise linear finite element method. A priori estimates and error bounds are established in the semi-discrete and the fully-discrete cases. The theoretical results are also illustrated in 1-D and 2-D. In the 1-D case solutions are typically periodic travelling waves, while in 2-D solutions may be spiral waves or 'target patterns'.



  3. Garvie M.R. and Trenchea C. (2007), "Finite element approximations of spatially extended predator-prey interactions with the Holling type II functional response," published in Numerische Mathematik, Vol. 107, pp. 641-667.

    We present the numerical analysis of two well-known reaction-diffusion systems modeling predator-prey interactions, where the local growth of prey is logistic and the predator displays the Holling type II functional response. Results are presented for two fully-practical piecewise linear finite element methods. We establish a priori estimates and error bounds for the semi-discrete and fully-discrete finite element approximations. Numerical results illustrating the theoretical results and spatiotemporal phenomena (e.g., spiral waves and chaos) are presented in 1-D and 2-D.



  4. Garvie M.R. (2007) , "Finite difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB," published in Bulletin of Mathematical Biology, Vol. 69. No. 3, pp. 931-956.

    We present two finite difference algorithms for studying the dynamics of spatially extended predator-prey interactions with the Holling Type II functional response and logistic growth of the prey. The algorithms are stable and convergent provided the time step is below a (non-restrictive) critical value. This is advantageous as it is well-known that the dynamics of nonlinear differential equations (DEs) can differ significantly from that of the underlying DEs themselves. This is particularly important for the spatially extended systems that are studied in this paper as they display a wide spectrum of ecologically relevant behavior, including chaos. Furthermore, there are implementational advantages of the methods. For example, due to the structure of the resulting linear systems, standard direct and iterative solvers are guaranteed to converge. Thus the algorithms are ideal for investigating the spatiotemporal dynamics of the solutions. We also present the results of numerical experiments in one and two space dimensions, and illustrate the simplicity of the numerical methods with short programs in MATLAB. Readers can download and edit the codes from PRED_PREY_SIM: Predator-Prey codes in Matlab.



  5. Garvie M.R. and Trenchea C. (2007) , "Optimal control of a 'nutrient-phytoplankton-zooplankton-fish' system," published in SIAM Journal On Control and Optimization, Vol. 46 No. 3, pp. 775-791.

    We consider the mathematical formulation, analysis, and numerical solution of an optimal control problem for a nonlinear `nutrient-phytoplankton-zooplankton-fish' reaction-diffusion system. We study the existence of optimal solutions, derive an optimality system, and determine optimal solutions. In the original spatially homogeneous formulation the dynamics of plankton were investigated as a function of parameters for nutrient levels and fish predation rate on zooplankton. In our paper the model is spatially extended and the parameter for fish predation treated as a control variable. The model has implications for the biomanipulation of food-webs in eutrophic lakes to help improve water quality. In order to illustrate the control of irregular spatiotemporal dynamics of plankton in the model we implement a semi-implicit (in time) finite element method with 'mass lumping', and present the results of numerical experiments in two space dimensions.



  6. Garvie M.R. and Golinski M. (2010) , "Metapopulation dynamics for spatially-extended predator-prey interactions," published in Ecological Complexity, Vol. 7 No. 2, pp. 55-59.

    Traditional metapopulation theory classifies a metapopulation as a spatially homogeneous population that persists on neighboring habitat patches. The fate of each population on a habitat patch is a function of a balance between births and deaths via establishment of new populations through migration to neighboring patches. In this study, we expand upon traditional metapopulation models by incorporating spatial heterogeneity into a previously studied two-patch nonlinear ordinary differential equation metapopulation model, in which the growth of a general prey species is logistic and growth of a general predator species displays a Holling type II functional response. The model described in this work assumes that migration by generalist predator and prey populations between habitat patches occurs via a migratory corridor. Thus, persistence of species is a function of local population dynamics and migration between spatially heterogeneous habitat patches. Numerical results generated by our model demonstrate that population densities exhibit periodic plane-wave phenomena, which appear to be functions of differences in migration rates between generalist predator and prey populations. We compare results generated from our model to results generated by similar, but less ecologically realistic work, and to observed population dynamics in natural metapopulations.



  7. Garvie M.R. and Trenchea C. (2010), "Spatiotemporal dynamics of two generic predator-prey models,"
    published in Journal of Biological Dynamics, Vol. 4 No. 6, pp. 559-570.

    We present the analysis of two reaction-diffusion systems modeling predator-prey interactions with the Holling Type II functional response and logistic growth of the prey. Initially we undertake the local analysis of the systems, deriving conditions on the parameters that guarantee a stable limit cycle in the reaction kinetics, and construct arbitrary large invariant regions in the equal diffusion coefficient case. We then provide an a priori estimate that leads to the global well-posedness of the classical (nonnegative) solutions, given any nonnegative L&infin- initial data. In order to verify the theoretical results numerical results are provided in two space dimensions using a Galerkin finite element method with piecewise linear continuous basis functions.



  8. Schwalb A.N., Garvie M.R. and Ackerman J.D. (2010), "Dispersion of freshwater mussel larvae in a lowland river," published in Limnology and Oceanography, Vol. 55 No. 2, pp. 628-638.

    We examined the dispersal of larvae (glochidia) of a common unionid mussel species Actinonaias ligamentina, which need to attach to a host fish in order to develop into juveniles, in a lowland river (Sydenham River, Ontario, Canada). Results from field trials are compared with the numerical solution of a 1D convection-diffusion partial differential equation model.



  9. Garvie M.R., Maini P.K. and Trenchea C. (2010), "An efficient and robust numerical algorithm for estimating parameters in Turing systems," published in Journal of Computational Physics, Vol. 229, pp. 7058-7071

    We present a new algorithm for estimating parameters in reaction-diffusion systems that display pattern formation via the mechanism of diffusion-driven instability. A Modified Discrete Optimal Control Algorithm (MDOCA) is illustrated with the Schnakenberg and Gierer-Meinhardt reaction-diffusion systems using PDE constrained optimization techniques. The MDOCA algorithm is a modification of a standard variable-step gradient algorithm that yields a huge saving in computational cost. The results of numerical experiments demonstrate that the algorithm accurately estimated key parameters associated with stationary target functions generated from the models themselves. Furthermore, the robustness of the algorithm was verified by performing experiments with target functions perturbed with various levels of additive noise. The MDOCA algorithm could have important applications in the mathematical modeling of realistic Turing systems when experimental data are available. A Modified Discrete Optimal Control Algorithm (MDOCA) is illustrated with the Schnakenberg and Gierer-Meinhardt reaction-diffusion systems using PDE constrained optimization techniques. The MDOCA algorithm is a modification of a standard variable-step gradient algorithm that yields a huge saving in computational cost. The results of numerical experiments demonstrate that the algorithm accurately estimated key parameters associated with stationary target functions generated from the models themselves. Furthermore, the robustness of the algorithm was verified by performing experiments with target functions perturbed with various levels of additive noise. The MDOCA algorithm could have important applications in the mathematical modeling of realistic Turing systems when experimental data are available.



  10. Garvie M.R. and Trenchea C. (2011), "A three level finite element approximation of a pattern formation model in developmental biology," published in Numerische Mathematik (online first DOI 10.1007/s00211-013-0591-z).

    This paper concerns a second-order, three level piecewise linear finite element scheme 2-SBDF [J. RUUTH, Implicit-explicit methods for reaction-diffusion problems in pattern formation, J. Math. Biol., 34 (1995), pp. 148-176] for approximating the stationary (Turing) patterns of a well- known experimental substrate-inhibition reaction-diffusion (‘Thomas’) system [D. THOMAS, Artificial enzyme membranes, transport, memory and oscillatory phenomena, in Analysis and control of immobilized enzyme systems, D. Thomas and J.P. Kernevez, eds., Springer, 1975, pp. 115-150]. A numerical analysis of the semi-discrete in time approximations leads to semi-discrete a priori bounds and an optimal error estimate. The analysis highlights the technical challenges in undertaking the numerical analysis of multi-level (≥ 3) schemes. We illustrate the effectiveness of the numerical method by repeating an important classical experiment in mathematical biology, namely, to approximate the Turing patterns of the Thomas system over a schematic mammal skin domain with fixed geometry at various scales. We also make some comments on the correct procedure for simulating Turing patterns in general reaction-diffusion systems.



  11. Garvie M.R. and Trenchea C. (2011), "Identification of space-time distributed parameters in the Gierer-Meinhardt reaction-diffusion system " SIAM Journal on Applied Mathematics, Vol. 74, No. 1, pp. 147-166

    We consider parameter identification for the classic Gierer- Meinhardt reaction-diffusion system. The original Gierer-Meinhardt model [A. Gierer and H. Meinhardt, Kybernetik, 12 (1972), pp. 30-39] was formulated with constant parameters and has been used as a prototype system for investigating pattern formation in developmental biology. In our paper the parameters are extended in time and space and used as distributed control variables. The methodology employs PDE-constrained optimization in the context of image-driven spatiotemporal pattern formation. We prove the existence of optimal solutions, derive an optimality system, and determine optimal solutions. The results of numerical experiments in 2D are presented using the finite element method, which illustrates the convergence of a variable-step gradient algorithm for finding the optimal parameters of the system. A practical target function was constructed for the optimal control algorithm corresponding to the actual image of a marine angelfish.



  12. Garvie M.R., Burkardt J. and Morgan J. (2015), "Simple Finite Element Methods for Approximating Predator-Prey Dynamics in Two Dimensions Using MATLAB " Bulletin of Mathematical Biology, Vol. 77. No. 3, pp. 548-578

    We describe simple finite element schemes for approximating spatially extended predator–prey dynamics with the Holling type II functional response and logistic growth of the prey. The finite element schemes generalize ‘Scheme 1’ in the paper by Garvie (Bull Math Biol 69(3):931–956, 2007). We present user-friendly, open-source MATLAB code for implementing the finite element methods on arbitrary-shaped two-dimensional domains with Dirichlet, Neumann, Robin, mixed Robin–Neumann, mixed Dirichlet–Neumann, and Periodic boundary conditions. Users can download, edit, and run the codes from http://www.uoguelph.ca/~mgarvie/. In addition to discussing the well posedness of the model equations, the results of numerical experiments are presented and demonstrate the crucial role that habitat shape, initial data, and the boundary conditions play in determining the spatiotemporal dynamics of predator–prey interactions. As most previous works on this problem have focussed on square domains with standard boundary conditions, our paper makes a significant contribution to the area.



  13. Garvie M.R. and Morgan J. and Sharma V. (2017), "Finite element approximation of a spatially structured metapopulation PDE model " Computers and Mathematics with Applications, Vol. 74. No. 5, pp. 934–947

    We present a new fully spatially structured PDE metapopulation model for predator–prey dynamics in d <= 3 space dimensions. A nonlinear reaction–diffusion system of Rosenzweig–MacArthur form models predator–prey dynamics in two ‘high’ quality patches embedded in a ‘low’ quality subdomain, where species can diffuse, convect and die. Our model substantially generalizes and improves earlier fully structured metapopulation models. After a nondimensionalization procedure, in order to approximate the metapopulation model we present a fully discrete Galerkin finite element method in two space dimensions, which is a generalization of the finite element method analyzed in a previous single patch predator–prey model. The numerical solutions are illustrated for some test cases using MATLAB. Numerical experiments demonstrate that the initial local extinction of predators in one patch leads to waves of recolonization from another patch. In an appendix we also give an outline for the proof of the well-posedness of the model.



  14. Diele F. and Garvie M.R. and Trenchea C. (2017), " Numerical analysis of a first-order in time implicit-symplectic scheme for predator-prey systems " Computers and Mathematics with Applications, Vol. 74. No. 5, pp. 948-961

    The numerical solution of reaction-diffusion systems modelling predator-prey dynamics using implicit-symplectic (IMSP) schemes is relatively new. When applied to problems with chaotic dynamics they perform well, both in terms of computational effort and accuracy. However, until the current paper, a rigorous numerical analysis was lacking. We analyse the semi-discrete in time approximations of a first-order IMSP scheme applied to spatially extended predator-prey systems. We rigorously establish semi-discrete a priori bounds that guarantee positive and stable solutions, and prove an optimal a priori error estimate. This analysis is an improvement on previous theoretical results using standard implicit-explicit (IMEX) schemes. The theoretical results are illustrated via numerical experiments in one and two space dimensions using fully-discrete finite element approximations.



  15. Submitted papers: (links not active)

    Conference proceedings:



  16. Trenchea C. and Garvie M.R. , "Biomanipulation of Food-Webs in Eutrophic Lakes.," published in Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, Louisiana, USA.

    This is an adaptation of the SIAM paper listed above for a conference presentation.



   
   
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Last updated: May 8, 2015 by Marcus R Garvie